Research report simulations
Thom Volker
September 20, 2021
1 Simulation 1
In this simulation, I created data according to a multivariate normal distribution, where all variables had variance \(\sigma^2 = 1\). The predictor variables \(X\) with dimensions \((n \times k)\) with \(n \in \{25, 50, 100, 200, 400, 800\}\) and \(k = 4\) were correlated with \(\rho = 0.3\). The outcome variable \(Y\), and the ratio of the regression coefficients equals \(\beta_4 = 4\beta_1, \beta_3 = 3\beta_1, \beta_2 = 2\beta_1\), with the exact size depending on the proportion explained variance \(R^2\) of the regression model.
The hypothesis of interest was \[ H_i: \beta_1 < \beta_2 < \beta_3 < \beta_4, \] and was evaluated against an unconstrained hypothesis \[ H_u: \beta_1, \beta_2, \beta_3, \beta_4. \]
output <- output %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output %>%
group_by(n, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output %>%
group_by(nsim, n, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n), y = log(bf_h1), fill = as.factor(n))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")
2 Simulation 2
Simulation 2 is basically a replication of simulation 1, with as the only difference that in all “meta-studies” there is a single study randomly selected to have a sample size of \(n = 25\). So again, the hypothesis of interest that is evaluated is \[ H_i: \beta_1 < \beta_2 < \beta_3 < \beta_4, \] relative to \[ H_u: \beta_1, \beta_2, \beta_3, \beta_4. \]
output <- output %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output %>%
group_by(n_initial, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n_initial, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n_initial, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n_initial, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output %>%
group_by(nsim, n_initial, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n_initial),
y = log(bf_h1),
fill = as.factor(n_initial))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n_initial, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n_initial, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n_initial, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")
3 Simulation 3
Simulation 3 is an extension of simulation 1 in the sense that the number of predictor variables equals \(k = 5\), bivariately correlated still with \(\rho = 0.3\). The ratio of the regression coefficients now equalled \(\beta_5 = 3\beta_1, \beta_4 = 2\beta_1, \beta_3 = \beta_2 = \beta_1\), and hence there are three regression coefficients of equal size.
The hypothesis of interest was \[ H_i: \beta_1 > 0; \beta_2 > 0; \beta_3 > 0, \]
evaluated against the unconstrained hypothesis \[ H_u: \beta_1, \beta_2, \beta_3, \beta_4, \beta_5. \] Hence, not every regression coefficient is included in the hypothesis of interest (the predictors with the largest size are not included in the hypothesis of interest).
output <- output %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output %>%
group_by(n, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output %>%
group_by(nsim, n, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n),
y = log(bf_h1),
fill = as.factor(n))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")
4 Simulation 4
Simulation 4 is exactly similar to simulation 3, except that now \(\beta_1, \beta_2, \beta_3\) are combined into a single variable that is the sum of the three separate variables, which yields the hypothesis of interest \[ H_i: \beta_{sum} > 0, \] evaluated against \[ H_u: \beta_{sum}, \beta_4, \beta_5. \]
output <- output %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output %>%
group_by(n, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output %>%
group_by(nsim, n, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n),
y = log(bf_h1),
fill = as.factor(n))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")
5 Simulation 5
Simulation 5 is exactly similar to simulations 3 and 4, except that now \(\beta_1, \beta_2, \beta_3\) are combined into a single variable that is the mean, rather than the sum, of the three separate variables. Hence, the hypothesis of interest was \[ H_i: \beta_{mean} > 0, \] evaluated against \[ H_u: \beta_{mean}, \beta_4, \beta_5. \]
output <- output %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output %>%
group_by(n, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output %>%
group_by(nsim, n, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n),
y = log(bf_h1),
fill = as.factor(n))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")
6 Simulation 6
In simulation 6, the set-up is fairly similar to simulation 1, except that now the number of predictors equals \(k = 2\), with a correlation of \(\rho = 0.3\) and a sample size of \(n \in \{50, 100, 200, 400, 800\}\). The ratio of regression coefficients here equals \(\beta_2 = 2\beta_1\). This situation is adjusted in a second scenario in which \(X_2\) is split up in three distinct categories with cut-offs \((-Inf,-0.431], (-0.431,0.431], (0.431, Inf]\), resulting in an \(\beta_{intercept}, \beta_{2D2}, \beta_{2D3}\). Since the smallest group \(n = 25\) resulted in complete separation in some datasets, I excluded this sample size.
6.1 Single continuous variable
output_c <- output_c %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output_c %>%
group_by(n, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output_c %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output_c %>%
group_by(nsim, n, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n),
y = log(bf_h1),
fill = as.factor(n))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output_c %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")
6.2 Continuous variable split into dummies
output_d <- output_d %>%
mutate(bf_u = map(fit, ~c(.x[,7], unconstrained = 1)),
bf_h1 = map_dbl(bf_u, ~.x[1]),
pmp_u = map(bf_u, ~.x / sum(.x)),
r2 = paste0("R^2: ", r2))
output_d %>%
group_by(n, r2, model) %>%
summarize(bf_h1 = mean(bf_h1), .groups = "drop") %>%
ggplot(mapping = aes(x = n, y = bf_h1, col = model)) +
geom_line() +
geom_hline(yintercept = 1, col = "grey", size = 1) +
scale_color_brewer(palette = "Set1") +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
labs(y = expression(BF["H"["i,u"]]),
x = "Sample size",
title = "Individual study Bayes Factor")
hypnames <- c("Hi", "Complement", "Unconstrained")
output_d %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0)) %>%
group_by(n, r2, model) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = model)) +
geom_line() +
facet_wrap(~r2, labeller = label_parsed) +
theme_minimal() +
scale_color_brewer(palette = "Set1") +
labs(y = "Hypothesis selection rates",
x = "Sample size",
title = "Individual study true hypothesis selection rate")
output_d %>%
group_by(nsim, n, r2) %>%
summarize(bf_h1 = prod(bf_h1), .groups = "drop") %>%
ggplot(aes(x = as.factor(n),
y = log(bf_h1),
fill = as.factor(n))) +
geom_boxplot() +
geom_hline(yintercept = 0, col = "grey", size = 1) +
facet_wrap(~r2, labeller = label_parsed) +
scale_fill_brewer(palette = "Set2") +
theme_minimal() +
labs(y = expression(log~BF["H"["i,u"]]),
x = "Sample size",
title = "Combined Bayes Factor") +
theme(legend.position = "none")
output_d %>%
mutate(bf_u = map(bf_u, ~as_tibble(matrix(.x,
ncol = 3,
dimnames = list(NULL,
hypnames))))) %>%
unnest(bf_u) %>%
group_by(nsim, n, r2) %>%
summarise(across(c(Hi, Complement, Unconstrained), prod), .groups = "drop") %>%
mutate(Hi = ifelse(Hi > Unconstrained, 1, 0),
Unconstrained = ifelse(Unconstrained > Hi, 1, 0)) %>%
group_by(n, r2) %>%
summarize(thr = mean(Hi), .groups = "drop") %>%
ggplot(aes(x = n, y = thr, col = r2)) +
geom_line() +
scale_color_brewer(palette = "Set1", labels = scales::parse_format()) +
theme_minimal() +
labs(y = "True hypothesis rate",
x = "Sample size",
title = "True hypothesis rate of hypothesis of interest versus unconstrained")